I'll keep silent about PUSH as I don't know how much Darren wants disclosed. Not that I know all of it, however he has shown me some of it and talked about a lot of the rest. In any case, it's his business (literally, huh?) how he does things.
About the DHXair. So the damper and the boost chamber are conneced by a little orifice which is kept closed by an oil metring valve called, you guessed it, the Boost Valve. Behind the valve is a piston which separates the oil from the boost chamber air. In addition, there's a piston inside of the damper which is shimmed bi-directionally, meaning, there are shims for compression and there are shims for rebound.
So, the damper is filled with oil, completely, there's no air there whatsoever. When you compress the shock, oil is being forced through the shimmed piston and the boost valve into the reservoir, however, the valve which meters the oil flow into the reservoir is being kept closed by the air pressure in the boost chamber. Eventually, the force of oil overcomes the force of air and oil begins to flow into the reservoir. Oil flowing into the reservoir is pushing down the piston which reduces the air volume in the boost chamber. This causes the air pressure to go up, which in turn exerts more force on the metering valve, wanting to close it. At the end of the stroke the force of air begins to overcome the force of oil, closing the metering valve and preventing harsh bottomout. Depending on you BO adjustment, the air chamber has a given size to begin with. If it is a smaller chamber (BO screwed in) the force of air rises faster, if the chamber size is larger, it raises slower. Ideally you would have it set so that it slows down the shaft speed at the end of the stroke enough to prevent a hard bottomout.
Propedal is simply a little coil spring which preloads the metering valve. It is a coil assist to the air pressure in the boost chamber. I have found that effectively, you can achieve the exact same pedaling characteristic by upping the pressure in the boost chamber slightly while gaining a smoother stroke, because the force of coil very quickly is overcome by the force air and oil and you can feel a little harshness at the transition.
Hope this helps. If anything isn't clear, let me know.
_MK